SCHWEITZER 1983 Problem 6

Let $T$ be a bounded linear operator on a Hilbert space $H$ , and assume that $\|T^n \| \leq 1$ for some natural number $n$ . Prove the existence of an invertible linear operator $A$ on $H$ such that $\| ATA^{-1} \| \leq 1$ .

E. Druszt